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Resonant eigenstates for a quantized chaotic system

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IOP PUBLISHING NONLINEARITY

Nonlinearity 20 (2007) 1387–1420 doi:10.1088/0951-7715/20/6/004

Resonant eigenstates for a quantized chaotic system

Stephane Nonnenmacher and Mathieu Rubin

Service de Physique Theorique, CEA/DSM/PhT, Unite de Recherche Associe CNRS,CEA/Saclay, 91191 Gif-sur-Yvette, France

E-mail: [email protected]

Received 10 October 2006, in final form 22 March 2007Published 3 May 2007Online at stacks.iop.org/Non/20/1387

Recommended by B Eckhardt

AbstractWe study the spectrum of quantized open maps as a model for the resonancespectrum of quantum scattering systems. We are particularly interested in openmaps admitting a fractal repeller. Using the ‘open baker’s map’ as an example,we numerically investigate the exponent appearing in the fractal Weyl law forthe density of resonances; we show that this exponent is not related with the‘information dimension’, but rather the Hausdorff dimension of the repeller.We then consider the semiclassical measures associated with the eigenstates:we prove that these measures are conditionally invariant with respect to theclassical dynamics. We then address the problem of classifying semiclassicalmeasures among conditionally invariant ones. For a solvable model, the ‘Walsh-quantized’ open baker’s map, we manage to exhibit a family of semiclassicalmeasures with simple self-similar properties.

Mathematics Subject Classification: 35B34, 37D20, 81Q50, 81U05

(Some figures in this article are in colour only in the electronic version)

1. Introduction

1.1. Quantum scattering on RD and resonances

In a typical scattering system, particles of positive energy come from infinity, interact with alocalized potential V (q) and then leave to infinity. The corresponding quantum HamiltonianHh = −h2� + V (q) has an absolutely continuous spectrum on the positive axis. Yet, Green’sfunction G(z; q ′, q) = 〈q ′|(Hh − z)−1|q〉 admits a meromorphic continuation from the upperhalf-plane {�z > 0} to (some part of) the lower half-plane {�z < 0}. This continuationgenerally has poles zj = Ej − i�j/2, �j > 0, which are called resonances of the scatteringsystem.

The probability density of the corresponding ‘eigenfunction’ ϕj (q) decays in time likee−t�j /h, so physically ϕj represents a metastable state with a decay rate �j/h or lifetime

0951-7715/07/061387+34$30.00 © 2007 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1387

http://dx.doi.org/10.1088/0951-7715/20/6/004

mailto: [email protected]

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1388 S Nonnenmacher and M Rubin

τj = h/�j . In the semiclassical limit h → 0, we will call ‘long-living’ the resonances zj suchthat �j = O(h), equivalently with lifetimes bounded away from zero.

The eigenfunctionϕj (q) is meaningful only near the interaction region, while its behaviouroutside that region (exponentially increasing outgoing waves) is clearly unphysical. As aresult, one practical method to compute resonances (at least approximately) consists of addinga smooth absorbing potential −iW(q) to the Hamiltonian H , thereby obtaining a nonself-adjoint operator HW,h = Hh − iW(q). The potential W(q) is supposed to vanish in theinteraction region but is positive outside: its effect is to absorb outgoing waves, as opposedto a real positive potential which would reflect the waves back into the interaction region.Equivalently, the (nonunitary) propagator e−iHW,h/h kills wavepackets localized oustide theinteraction region.

The spectrum of HW,h in some neighbourhood of the positive axis is then made upof discrete eigenvalues zj associated with square-integrable eigenfunctions ϕj . AbsorbingHamiltonians of the type ofHW,h have been widely used in quantum chemistry to study reactionor dissociation dynamics [23, 40]; in these works it is implicitly assumed that eigenvalues zjclose to the real axis are small perturbations of the resonances zj and that the correspondingeigenfunctions ϕj (q), ϕj (q) are close to one another in the interaction region. Very close tothe real axis (namely, for |�zj | = O(hn) with n sufficiently large), one can prove that thisis indeed the case [43]. Such very long-living resonances are possible when the classicaldynamics admits a trapped region of positive Liouville volume. In that case, resonances andthe associated eigenfunctions can be approximated by quasimodes of an associated closedsystem [44].

1.2. Resonances in chaotic scattering

We are interested in a different situation, where the set of trapped trajectories has volumezero and is a fractal hyperbolic repeller. This case encompasses the famous 3-disc scattererin two dimensions [13] or its smoothing, namely, the 3-bump potential introduced in [41] andnumerically studied in [24]. Resonances then lie deeper below the real line (typically, �j � h)and are not perturbations of an associated real spectrum. Previous studies have focused oncounting the number of resonances in small discs around the energy E, in the semiclassicalregime. Based on the seminal work of Sjostrand [41], several authors have conjectured thefollowing Weyl-type law:

#{zj ∈ Res(Hh) : |E − zj | � γ h

} ∼ C(γ )h−d . (1.1)

Here the exponent d is related to the trapped set at energy E: the latter has the (Minkowski)dimension 2d + 1. This asymptotics was numerically checked by Lin and Zworski for the3-bump potential [24, 26] and by Guillope, Lin and Zworski for scattering on a hyperbolicsurface [15]. However, only the upper bounds for the number of resonances could be rigorouslyproven [15, 41, 42, 49].

To avoid the complexity of ‘realistic’ scattering systems, one can study simpler models,namely, quantized open maps on a compact phase space, for instance, the quantized openbaker’s map studied in [20, 30, 31] (see section 2.1). Such a model is meant to mimic thepropagator of the nonself-adjoint Hamiltonian HW,h in the case where the classical flow atenergyE is chaotic in the interacting region. The above fractal Weyl law has a direct counterpartin this setting; such a fractal scaling was checked for the open kicked rotator in [38] and forthe ‘symmetric’ baker’s map in [30]. In section 4.1 we numerically check this fractal law foran asymmetric version of the open baker’s map; apart from extending the results of [30], thismodel allows us to specify more precisely the dimension d appearing in the scaling law. To our

Resonant eigenstates for a quantized chaotic system 1389

knowledge, so far the only system for which the asymptotics corresponding to (1.1) could berigorously proven is the ‘Walsh-quantization’ of the symmetric open baker’s map [31], whichwill be described in section 6.

After counting resonances, the next step consists of studying the long-living resonanteigenstates ϕj or ϕj . Some results on this matter have been announced by Zworski andthe first author in [32]. Interesting numerics were performed by Lebental and coworkersfor a model of an open stadium billiard, relevant for describing an experimental micro-lasercavity [22]. In the framework of quantum open maps, eigenstates of the open Chirikov maphave been numerically studied by Casati et al [5]; the authors showed that, in the semiclassicallimit, the long-living eigenstates concentrate on the classical hyperbolic repeller. They alsofound that the phase space structure of the eigenstates is much correlated with their decayrate. In this paper we will consider general ‘quantizable’ open maps and formalize the aboveobservations into rigorous statements. Our main result is theorem 1 (see section 5.1), whichshows that the semiclassical measure associated with a sequence of quantum eigenstates is(up to subtleties due to discontinuities) necessarily an eigenmeasure of the classical open map.Such eigenmeasures are necessarily supported on the backward trapped set, which, in the caseof a chaotic dynamics, is a fractal subset of the phase space: this motivated the denominationof ‘quantum fractal eigenstates’ used in [5]. Let us mention that eigenstates of quantized openmaps have been studied in parallel by Keating and coworkers [20]. Our theorem 1 provides arigorous version of statements contained in their work.

Inspired by our experience with closed chaotic systems, in section 5.2 we attemptto classify semiclassical measures among all possible eigenmeasures, in particular for theopen baker’s map. In the case of the ‘standard’ quantized open baker, the classificationremains open. In section 6 we consider a solvable model, the Walsh-quantization of theopen baker’s map, introduced in [30, 31]. For that model, one can explicitly constructsome semiclassical measures and partially answer the above questions. A further study ofsemiclassical measures for the Walsh model will appear in a joint publication with Keating,Novaes and Sieber.

2. The open baker’s map

2.1. Closed and open symplectic maps

Although many of the results we will present deal with a particular family of maps on the2-torus, namely the family of baker’s maps, we start by some general considerations on openmaps defined on a compact metric space M , equipped with a probability measure µL. Weborrow some ideas and notation from the recent review of Demers and Young [9]. We startwith an invertible map T : M → M , which we assume to be piecewise smooth and to preservethe measure µL (when M is a symplectic manifold, µL is the Liouville measure). We then diga hole in M , which is a certain subset H ⊂ M , and decide that points falling in the hole areno more iterated but rather ‘disappear’ or ‘go to infinity’. The hole is assumed to be a Borelsubset of M .

Taking Mdef= M \H , we are thus lead to consider the open map T = T|M : M → M or,

equivalently say that T sends points in the hole to infinity. By iterating T , we see that any pointx ∈ M has a certain time of escape n(x), which is the smallest integer such that T n(x) ∈ H(this time can be infinite). For each n ∈ N∗ we consider

Mn = {x ∈ M, n(x) � n} =n−1⋂j=0

T −j (M) . (2.1)


This is the domain of definition of the iterated map T n. The forward trapped set for the openmap T is made up of the points which will never escape in the future:

�− =⋂n�1

Mn =∞⋂j=0

T −j (M) . (2.2)

These definitions allow us to split the full phase space into a disjoint union

M =( ∞⊔n=1

Mn

)� �−, (2.3)

where M1def= H , and for each n � 2, Mn

def= Mn−1 \Mn is the set of points escaping exactlyat time n.

We also consider the backward evolution given by the inverse map T −1. The holefor this backwards map is the set H−1 = T (H), and we call T −1 the restriction of T −1

to M−1 = T (M) (the ‘backwards open map’). We also define M−n = ⋂nj=1 T

j (M),M−1 = H−1, M−n = M−n+1 \M−n (n � 2). This leads to the the backward trapped set

�+ =∞⋂j=1

T j (M) and the trapped set K = �− ∩ �+ . (2.4)

We also have a ‘backward partition’ of the phase space:

M =( ∞⊔n=1

M−n

)� �+. (2.5)

The dynamics of the open map T is interesting if the trapped sets are not empty. This is thecase for the open baker’s map we will study more explicitly (see section 2.2.2 and figure 2).

In order to quantize the maps T and T , one needs further assumptions. In general, M is asymplectic manifold, µL its Liouville measure and T a canonical transformation on M . Also,we will assume that T is sufficiently regular (see section 3.1).

Yet, one may also consider ‘quantizations’ on more general phase spaces, such as in theaxiomatic framework of Marklof and O’Keefe [28]. As we explain in section 6.1.2, the ‘Walsh-quantization’ of the baker’s map is easier to analyse if we consider it as the quantization of anopen map on a certain symbolic space, which is not a symplectic manifold. By extension, wealso call ‘Liouville’ the measure µL for this case.

2.2. The open baker’s map and its symbolic dynamics

We present the closed and open maps which will be our central examples: the baker’s mapsand their associated symbolic shifts.

2.2.1. The closed baker. The phase space of the baker’s map is the two-dimensional torusM = T2 [0, 1) × [0, 1). A point on T2 is described with the coordinates x = (q, p),which we call, respectively, the position (horizontal) and the momentum (vertical), to insiston the symplectic structure dq ∧ dp. We split T2 into three vertical rectangles Ri with widths

(r0, r1, r2)def= r (such that r0 + r1 + r2 = 1, with all rε > 0) and first define a closed baker’s


Figure 1. Sketch of the closed baker’s map Br and its open counterpart Br, for the caser = rsym = (1/3, 1/3, 1/3). The three rectangles form a Markov partition.

map on T2 (see figure 1):

(q, p) �→ Br(q, p)def= (q ′, p′) =

(q

r0, p r0

)if 0 � q < r0,(

q − r0

r1, p r1 + r0

)if r0 � q < r0 + r1,(

q − r0 − r1

r2, p r2 + r1 + r0

)if 1 − r2 � q < 1 .

(2.6)

This map is invertible and symplectic on T2. It is discontinuous on the boundaries of therectanglesRi but smooth (actually, affine) inside them. We now recall how Br can be conjugatedwith a symbolic dynamics. We introduce the right shift σ on the symbolic space� = {0, 1, 2}Z

ε = . . . ε−2ε−1 · ε0ε1 . . . ∈ � �−→ σ (ε) = . . . ε−2ε−1ε0 · ε1 . . . .

The symbolic space � can be mapped to the 2-torus as follows: every bi-infinite sequenceε ∈ � is mapped to the point x = Jr(ε) with coordinates

q(ε) =∞∑k=0

rε0rε1 · · · rεk−1 αεk , p(ε) =∞∑k=1

rε−1rε−2 · · · rε−k+1 αε−k , (2.7)

where we have set α0 = 0, α1 = r0, α2 = r0 + r1. The position coordinate (unstabledirection) depends on symbols on the right of the comma, while the momentum coordinate(stable direction) depends on symbols on the left. The map Jr : � → T2 is surjective but notinjective; for instance, the sequences . . . εn−1εn000 . . . (with εn �= 0) and . . . εn−1(εn−1)222 . . .have the same image on T2. For this reason, it is convenient to restrict Jr to the subset�′ ⊂ �

obtained by removing from� the sequences ending by . . . 2222 . . . on the left or the right. �′

is invariant through the shift σ . The map Jr|�′ : �′ → T2 is now bijective, and it conjugatesσ|�′ with the baker’s map Br on T2:

Br = Jr|�′ ◦ σ ◦ (Jr|�′)−1 . (2.8)

Any finite sequence ε = ε−m . . . ε−1 · ε0 . . . εn−1 represents a cylinder [ε] ⊂ �, which consistsof the sequences sharing the same symbols between indices −m and n− 1. We call Jr([ε]) arectangle on the torus (sometimes we will note it as [ε]). This rectangle has sides parallel tothe two axes; it has width rε0rε1 · · · rεn−1 and height rε−1 · · · rε−m .

For any triple, the Liouville measure on T2 is the push-forward through Jr of a certainBernoulli measure on �, namely, µ�L = ν

�−r × ν�+

r (see section 2.3.5).


p

0

1

1q

Figure 2. In black we approximate the forward (left), backward (centre) and joint (right) trappedsets for the symmetric open baker Brsym . On the left (respectively, centre), red/grey scales (fromwhite to dark) correspond to points escaping at successive times in the future (respectively, past),that is, to sets Mn (respectively, M−n), for n = 1, 2, 3, 4.

2.2.2. The open baker. We choose to take for the hole the middle rectangle H = R1 ={r0 � q < 1 − r2}. We thus obtain an ‘open baker’s map’ Br defined on M = T2 \H :

(q, p) �→ Br(q, p)def= (q ′, p′) =

(q

r0, p r0

)if 0 � q < r0,

∞ if r0 � q < 1 − r2,(q − 1 + r2

r2, p r2 + 1 − r2

)if 1 − r2 � q < 1 .

The ‘inverse map’ B−1r is defined on the set M−1 = Br(M), which is outside the backward

hole H−1 = Br(H) = {r0 � p < 1 − r2} (see figure 1, right).Due to the choice of the hole, this open map is still easy to analyse through symbolic

dynamics: the holeR1 is the image through Jr of the set {ε0 = 1}∩�′. Let us define as followsthe open shift σ on �:

ε ∈ � �−→ σ(ε)def={

∞ if ε0 = 1,

σ (ε) if ε0 ∈ {0, 2} . (2.9)

Jr|�′ conjugates the open shift σ|�′ with Br as in (2.8). Similarly, the backward open shiftσ−1, which kills the sequences s.t. ε−1 = 1 and otherwise moves the comma to the left, isconjugated with B−1

r .These conjugations allow us to easily characterize the various trapped sets of Br. On

the symbolic space �, the forward (respectively, backward) trapped set of the open shift σis given by the sequences ε such that εn ∈ {0, 2} for all n � 0 (respectively, for all n < 0).To obtain the trapped sets for the baker’s map, we restrict ourselves on �′ and conjugate byJr|�′ . The image sets are given by the direct products �− = Cr × [0, 1), �+ = [0, 1)× Cr andK = Cr ×Cr, where Cr is (up to a countable set) the Cantor set on [0, 1) adapted to the partitionr (see figure 2).

For future use, we define the subset �′′ ⊂ � by

�′′ def= � \ ({. . . 222 · 0ε1ε2 . . .} ∪ {. . . 222 · 2ε1ε2 . . .} ∪ {. . . ε−2ε−1 · ε0222 . . .}s) . (2.10)

We note that �′′ � �′ and that Jr(� \ �′′) is a subset of the discontinuity set of the map Br

(see section 3.2.2 and figure 4). The map Jr realizes a kind of semiconjugacy between σ|�′′

and B:

∀ε ∈ �′′, Jr ◦ σ(ε) = Br ◦ Jr(ε). (2.11)

It is understood above that both sides are ‘sent to infinity’ if ε0 = 1.


2.3. Eigenmeasures of open maps

Before defining eigenmeasures of open maps, we briefly recall how invariant measures emergein the study of the quantized closed maps.

2.3.1. Quantum ergodicity for closed chaotic systems. The quantum-classicalcorrespondence between a closed symplectic map T and its quantization TN (see section 3.1)has one important consequence: in the semiclassical limit N → ∞, stationary states of thequantum system (that is, eigenstates of TN ) should reflect the stationary properties of theclassical map, namely, its invariant measures. To be more precise, with any sequence ofeigenstates (ψN)N→∞ of the quantum map, one can associate at least one semiclassical measure(see section 3.1.2). Omitting problems due to the discontinuities of T , the quantum-classicalcorrespondence implies that a semiclassical measure µ must be invariant w.r.t. T :

T ∗ µ = µ ⇐⇒ for any Borel set S ⊂ T2, µ(T −1(S)) = µ(S) . (2.12)

µ is then also invariant w.r.t. the inverse map T −1.If the map T is ergodic w.r.t. the Liouville measure µL on M , the quantum ergodicity

theorem (or Schnirelman’s theorem [37]) states that, for ‘almost any’ sequence (ψN)N→∞,there is a unique associated semiclassical measure, which is µL itself.

Such a theorem was first proven for eigenstates of the Laplacian on compact Riemannianmanifolds with ergodic geodesic flow [7, 46] and then for more general Hamiltonians [17],billiards [14,48] and maps [4,47]. Quantum ergodicity for piecewise smooth maps was provenin a general setting in [28], and the particular case of the baker’s map was treated in [8]. Finally,in [1] it was shown that the (closed) Walsh–baker’s map, seen as the quantization of the rightshift σ on (�,µL), also satisfies quantum ergodicity with respect to µL.

It is generally unknown whether there exist ‘exceptional sequences’ of eigenstates,converging to a different invariant measure. The absence of such sequences is expressedby the quantum unique ergodicity conjecture [33], which has been proven only forsystems with arithmetic properties [21, 25]. This conjecture has been disproved for somespecific systems enjoying large spectral degeneracies at the quantum level, allowing forsufficient freedom to build up partially localized eigenstates [1, 12, 18]. Some specialeigenstates of the standard quantum baker with interesting multifractal properties havebeen numerically identified [29], but their persistence in the semiclassical limit remainsunclear.

2.3.2. Eigenmeasures of open maps. We now dig the holeH = M\M and consider the openmap T = T|M . Eigenmeasures (also called conditionally invariant measures) of maps ‘withholes’ have been less studied than their invariant counterparts. The recent paper of Demersand Young [9] summarizes most of the properties of these measures, for maps enjoying variousdynamical properties. Below we describe some of these properties for general maps, beforebeing more specific in the case of the baker’s map.

A probability measure µ on M which is invariant through T up to a multiplicative factorwill be called an eigenmeasure of T :

T ∗ µ = �µ µ ⇐⇒ for any Borel set S ⊂ M, µ(T −1(S)) = �µ µ(S) . (2.13)

Here �µ ∈ [0, 1] (or rather γµ = − log�µ) is called the ‘escape rate’ or the ‘decay rate’ ofthe eigenmeasure µ and is given by �µ = µ(M). It corresponds to the fact that a fractionof the particles in the support of µ escape at each step. Our definition slightly differs fromthe one in [9]: their measures are supported on M and normalized there, while we choose thenormalization µ(M) = 1.


Figure 3. Various components �(n)+ of the forward trapped set for the open baker Brsym . Variousred/grey scales (from light to dark) correspond to n = 1, 2, 3, 4.

Here are some simple properties.

Proposition 1. Let µ be an eigenmeasure of T with the decay rate �µ.If �µ = 0, µ is supported in the hole H .If �µ = 1, µ is supported in the trapped set K (invariant measure).If 0 < �µ < 1, µ is supported on the set �+\K .

Following the decomposition (2.3), the set �+ \K can be split into a disjoint union (seefigure 3):

�+\K =⊔n�1

�(n)+ , where �(n)+def= �+ ∩Mn = T −n+1�(1)+ , n � 1 .

Following [9, theorem. 3.1], any eigenmeasure can be constructed as follows.

Proposition 2. Take some � ∈ [0, 1) and ν an arbitrary Borel probability measure on�(1)+ = �+ ∩H . Define the probability measure µ on T2 as follows:

µ = (1 −�)∑n�0

�n (T ∗)n ν . (2.14)

Then T ∗ µ = �µ. All �-eigenmeasures of T can be written this way.

This construction shows that, as long as �(1)+ is not empty (that is, there exists at least apoint x0 trapped in the past but escaping in the future), there are plenty of eigenmeasures.

The case where the map T is uniformly hyperbolic has been studied in detail byChernov and Markarian [6, 27]. The set �(1)+ is then uncountable, and it is foliated by theunstable foliation. These authors focused on eigenmeasures of the open map T which areabsolutely continuous along the unstable direction. Even with this condition, one has plentyof eigenmeasures (we recall that for a closed hyperbolic map, unstable absolute continuity issatisfied for a unique invariant measure, namely, the SRB measure).

2.3.3. Pure point eigenmeasures. Applying the recipe of proposition 2 to the Dirac measureδx0 on an arbitrary point x0 ∈ �(1)+ , we obtain a simple, pure point�-eigenmeasure supported on


the backward trajectory (x−n = T −n(x0))n�0. For any � ∈ (0, 1), we call this eigenmeasure

µx0,�def= (1 −�)

∑n�0

�n δx−n . (2.15)

We recall that the pure point invariant measures of T are localized on periodic orbits of Ton K (which, for the case of a horseshoe map T|K , form a countable family). In contrast, foreach � ∈ [0, 1) the family of pure point eigenmeasures µx0,� is labelled by all x0 ∈ �(1)+ (anuncountable set).

2.3.4. Natural eigenmeasure. As discussed in [9], one may search for the definition of anatural eigenmeasure of T . The simplest (‘ideal’) definition reads as follows: for any initialmeasure ρ absolutely continuous w.r.t. µL, and such that ρ(�−) > 0,

µnat = limn→∞

T ∗n ρ‖T ∗n ρ‖ , where ‖µ‖ def= µ(M) . (2.16)

It was shown in [6, 27] that, for a hyperbolic open map, the limit exists and is independentof ρ. Besides, the natural measure is then absolutely continuous along the unstable direction.Yet, for a general open map the above limit does not necessarily exist or it may depend on theinitial distribution ρ [9].

In case (2.16) holds, the eigenvalue �nat = µnat(M) associated with this measure isgenerally called ‘the decay rate of the system’ by physicists. For a closed chaotic map T , thenatural measure is the Liouville measure µL (indeed, T ∗nρ → µL is equivalent to the factthat T is mixing with respect to µL). As explained in section 2.3.1, the quantum ergodicitytheorem shows that this particular invariant measure is ‘favoured’ by quantum mechanics. Oneinteresting question we will address is the relevance of µnat with respect to the quantized openmap TN .

2.3.5. Bernoulli eigenmeasures of the open baker. We now focus on the open baker Br andthe open shift σ it is conjugated with and construct a family of eigenmeasures called Bernoullieigenmeasures. Some of these measures will appear in section 6 as semiclassical measures forthe Walsh-quantized open baker.

The first equality in (2.7) maps the set �+ of one-sided sequences ·ε0ε1 . . . to the positioninterval [0, 1]. By a slight abuse, we also call this mapping Jr. We will first construct Bernoullimeasures on the symbol spaces �+ and � and then push them on the interval or the torususing Jr.

Let us recall the definition of Bernoulli measures on �+. Choose a weight distributionP = (P0, P1, P2), where Pε ∈ [0, 1] and P0 + P1 + P2 = 1. We define the measure ν�+

P on �+

as follows: for any n-sequence ε = ·ε0ε1 · · · εn−1, the weight of the cylinder [ε] is given by

ν�+P ([ε]) = Pε0 · · ·Pεn−1 .

The push-forward on [0, 1] of this measure will also be called a Bernoulli measure and calledνr,P = J ∗

r ν�+P . If we take Pε = rε for ε = 0, 1, 2, we recover νr,r = νLeb as the Lebesgue

measure on [0, 1]. If for some ε ∈ {0, 1, 2} we take Pε = 1, we get for νr,P the Diracmeasure at the point q(·εεε . . .), which takes the respective values 0, r0/(1 − r1) and 1. Forany other distribution P, the Bernoulli measure νr,P is purely singular continuous w.r.t. theLebesgue measure. Fractal properties of the measures νr,P were studied in [16]. If the weightPε vanishes for a single ε (e.g. P1 = 0), νr,P is supported on a Cantor set (e.g. Cr).

A Bernoulli measure on ν�−P can be defined similarly. By taking products of two Bernoulli

measures, one easily constructs eigenmeasures of the open shift σ or the open baker Br.


Proposition 3. Take any weight distribution P such that P1 < 1. Then the following hold.

(i) There is a unique auxiliary distribution, namely, P∗ = {P0/(P0 + P2), 0, P2/(P0 + P2)},such that the product measure

µ�Pdef= ν

�+P × ν

�−P∗

is an eigenmeasure of the open shift σ . The corresponding decay rate is

�P = 1 − P1 = P0 + P2 . (2.17)

(ii) For any r, the push-forward µr,P = J ∗r µ

�P is an eigenmeasure of the open baker Br.

By definition, the product measure has the following weight on a cylinder [ε] =[ε−m . . . εn−1]

µ�P ([ε]) = P ∗ε−m . . . P

∗ε−1Pε0 · · ·Pεn−1 .

The proof of the first statement is straightforward. There is a slight subtlety concerning push-forwards of σ -eigenmeasures, which is resolved in the following proposition.

Proposition 4. Let µ� be an eigenmeasure of the open shift σ : � → �. If µ� does notcharge the subset �\�′′ described in (2.10), that is if µ�(�\�′′) = 0, then its push-forwardµ = J ∗

r µ� on T2 is an eigenmeasure of Br.

Proof. Take any Borel set S ∈ T2. Then the following identities hold:

µ(B−1r (S))

def= µ�(J−1r ◦ B−1

r (S)) = µ�(σ−1 ◦ J−1r (S)) = �µ�(J−1

r (S))def= �µ(S) .

The second equality is a consequence of the semiconjugacy (2.11), which implies the fact thatthe symmetric difference of the sets J−1

r ◦ B−1r (S) and σ−1 ◦ J−1

r (S) is necessarily a subsetof �\�′′.

The conditionµ�(�\�′′) = 0 in the proposition cannot be removed. Indeed, by applyingthe construction of proposition 2 to an initial ‘seed’ supported on {. . . 222 · 1ε1 . . .}, one obtainsan eigenmeasure of σ charging �′′ and such that its push-forward is not an eigenmeasureof Br.

To prove the second point of the proposition, we remark that the only Bernoullieigenmeasure µ�P charging �\�′′ is the Dirac measure on the sequence . . . 2222 . . ., which ispushed forward to the delta measure at the origin of T2. �

If we take P = r, the push-forward is the natural eigenmeasure µr,r = µnat of the openbaker Br. If P �= P′, the Bernoulli eigenmeasures µ�P and µ�P′ are mutually singular (thereexists disjoint Borel subsets A, A′ of � such that νr,P(A) = νr,P′(A′) = 1), even thoughthey may share the same decay rate. Except in the case P = (1, 0, 0), P′ = (0, 0, 1), thepush-forwards µr,P and µr,P′ are also mutually singular.

3. Quantized open maps

3.1. ‘Axioms’ of quantization

For appropriate 2D-dimensional compact symplectic manifolds M , one may define a sequenceof ‘quantum’ Hilbert spaces (HN)N→∞ of finite dimensions N , which are related to Planck’sconstant by N ∼ h−D . Quantum states are normalized vectors in HN . One also wants toquantize observables, that is, functions a ∈ C∞(M), into operators OpN(a) on HN . Aninvertible (respectively, open) map T (respectively, T ) is quantized into a family of unitary


(respectively, contracting) operators TN (respectively, TN ), which satisfies certain propertieswhen N → ∞ (see below).

For the example that we will treat explicitly (the open baker’s map), the propagatorsof the closed and open maps are related by TN = TN ◦ �M,N , where �M,N is a projectorassociated with the subset M: it kills the quantum states microlocalized in the hole, whilekeeping unchanged the states microlocalized insideM [35]. Yet, in general the propagator TNcan be defined without having to construct TN beforehand.

The proof of our main result, theorem 1, only uses some ‘minimal’ properties of thequantized observables and maps. These properties were presented as ‘quantization axioms’by Marklof and O’Keefe in the case of closed maps [28]. We adopt the same approach,namely, define quantization through these ‘minimal axioms’ and state our result in this generalframework. Afterwards, we will check that these axioms are satisfied for the quantized openbaker.

3.1.1. Axioms on observables. All axioms will describe properties of the quantum operatorsin the semiclassical limit N → ∞. With a slight abuse of notation, we write AN ∼ BN whentwo families of operators (AN), (BN) on HN satisfy

‖AN − BN‖L(HN )N→∞−→ 0 .

The axioms concerning the quantization of observables read as follows [28, axiom 2.1].

Axioms 1. For any a ∈ C∞(M), the operators OpN(a) on HN must satisfy, in the limit N →∞:

OpN(a) ∼ OpN(a)† (asymptotic hermiticity),

OpN(a)OpN(b) ∼ OpN(ab) (0th order symbolic calculus),

limN→∞

N−1 Tr OpN(a) =∫M

a dµL (normalization) .

(3.1)

These axioms are satisfied by all standard quantization recipes (e.g. geometric or Toeplitzquantization on Kahler manifolds [47]). Note that they do not involve the symplectic structureon M and can thus be extended to more general phase spaces.

3.1.2. Semiclassical measures. From this, we may define semiclassical measures associatedwith sequences of (normalized) quantum states (ψN ∈ HN)N→∞. Such a sequence is said toconverge to the distribution µ on M iff, for any observable a ∈ C∞(M),

〈ψN, OpN(a)ψN 〉 N→∞−→∫M

a dµ . (3.2)

From the above axioms, one can show thatµ is necessarily a probability measure on M , whichis called the semiclassical measure associated with (ψN ∈ HN). By weak compactness, fromany sequence (ψN ∈ HN) one can always extract a subsequence (ψNk ) converging in the abovesense to a certain measure. The latter is called a semiclassical measure of the sequence (ψN).

3.1.3. Axioms on maps. In the axiomatic framework of [28], the conditions satisfied bythe unitary propagators (TN ) quantizing a closed map T consist of some form of quantum-classical correspondence (Egorov’s theorem), when evolving quantum observables OpN(a)through these propagators. However, if T is discontinuous (or nonsmooth) at some points, itspropagator T will exhibit diffraction phenomena around these ‘singular’ points, which alter thepropagation properties. At the classical level, a smooth observable a ∈ C∞(M) is transformed


into a smooth observable a ◦ T ∈ C∞(M) only if a vanishes near the singular points of T −1.As a result, the quantum-classical correspondence is a reasonable axiom only if the observablea is supported ‘far away’ from the singular set of T −1.

We now specify our assumptions on the open map T : M → M . Firstly, the hole H willbe a ‘nice’ set, that is, a set with nonempty interior and such that ∂H has Minkowski contentzero (i.e. the volume of its ε-neighbourhood vanishes when ε → 0). We also assume thatM , the domain of definition of T , can be decomposed using finitely or countably many openconnected sets Oi : M = ∪i�1Oi , with Oi ∩ Oj = ∅, and for each i, T|Oi : Oi → T (Oi) isa smooth canonical diffeomorphism, with all derivatives uniformly bounded. Let us split thehole into H = H � DH . The continuity set (respectively, discontinuity set) of the map T isdefined as

C(T ) = �i�1Oi ⊂ M , respectively, D(T )def= (M\C(T )) �DH .

We assume that D(T ) has Minkowski content zero. We have the decomposition M =C(T ) � H �D(T ). A similar decomposition holds for the inverse map:

M = C(T −1) � H−1 �D(T −1), with C(T −1) = T (C(T )) . (3.3)

Adapting the axioms of [28] to the case of open maps, we set as follows the characteristicproperty of the operators (TN)N→∞ quantizing T .

Axioms 2. We say that the operators (TN ∈ L(HN))N→∞ quantize the open map T iff

• for N large enough, ‖TN‖L(HN ) � 1,

• for any observable a ∈ C∞c (C(T

−1) � H−1), we have in the limit N → ∞

T†N OpN(a) TN ∼ OpN1M × (a ◦ T )) . (3.4)

Here C∞c (S) indicates the smooth functions compactly supported inside S and 1M is the

characteristic function on M .

Note that, if a is supported insideC(T −1)�H−1

, the function 1M ×(a◦T ) is well defined,smooth and supported insideC(T ). The factor 1M ensures that an observable supported insideH−1 is ‘semiclassically killed’ by the evolution through TN . The condition (3.4) reminds usof the definition of a ‘quantized weighted relation’ introduced in [31], but it is less precise (itonly describes the lowest order in h).

Remark 1. Going back to the problem of potential scattering mentioned in the introduction, weexpect the operators TN to share some spectral properties with the propagator exp(−iHW,h/h)of the ‘absorbing Hamiltonian’ in the semiclassical limit. The eigenvalues {λj } of TN should becompared with {e−izj /h} or with {e−izj /h}, where {zj } (respectively, {zj }) are the eigenvalues ofHW,h (respectively, the resonances ofHh). Similarly, the eigenstates of TN should share somemicrolocal properties with the eigenfunctions ϕj of HW,h (respectively, the resonant states ϕjof Hh) inside the interaction region. Accordingly, we will sometimes call ‘resonances’ and‘resonant eigenstates’ the eigenvalues/states of quantized open maps.

3.2. The quantum open baker’s map

For any dimension N = (2πh)−1, the quantum Hilbert space HN adapted to the torus phasespace is spanned by the orthonormal position basis {qj , j = 0, . . . , N − 1}, localized at thediscrete positions qj = j/N .

There are several standard ways to quantize observables on T2: Weyl quantization, Toeplitz(or anti-Wick) quantizations [4] or Walsh-quantization [1]. Weyl and anti-Wick quantizations


are equivalent to each other in the semiclassical limit, in the sense that OpWN (a) ∼ OpAWN (a) forany smooth observable. In contrast, the Walsh-quantization (see section 6.1) is not equivalentto the previous ones.

After recalling the definition of the anti-Wick quantization on T2 and the ‘standard’quantization of the baker’s map, following the original approach of Balazs–Voros, Saraceno,Saraceno–Vallejos [2, 34, 35], we check that the latter satisfies axiom 2 with respect to theanti-Wick quantization.

3.2.1. Anti-Wick quantization of observables. We recall the definition and properties ofcoherent states on T2, which we use to construct the anti-Wick quantization of observablesand by duality the Husimi representation of quantum states [4]. The Gaussian coherent statein L2(R), localized at the phase space point x = (q0, p0) ∈ R2 and with squeezing parameters > 0, is defined by the normalized wavefunction

�x,s(q)def=( s

πh

)1/4e−i p0q0

2h ei p0qh e−s (q−q0)

2

2h . (3.5)

When h = (2πN)−1, this state can be periodized on the torus, to yield the torus coherent stateψx,s ∈ HN with the following components on the basis of {qj }:

〈qj , ψx,s〉 = 1√N

∑ν∈Z

�x,s(j/N + ν), j = 0, . . . , N − 1 . (3.6)

For s > 0 fixed, these states are asymptotically normalized when N → ∞.With any squeezing s > 0 and inverse Planck’s constantN ∈ N we associate the anti-Wick

(or Toeplitz) quantization

f ∈ C∞(T2) �−→ OpAW,sN (f )

def=∫

T2|ψx,s〉〈ψx,s | f (x) N dx , (3.7)

which satisfies axiom 1 [4]. By duality, this quantization defines, for any state ψ ∈ HN , aHusimi distribution Hs

ψ :

∀f ∈ C∞(T2), H sψ(f )

def= 〈ψ,OpAW,sN (f )ψ〉 .

For ‖ψ‖ = 1, this distribution is a probability measure, with density given by the (smooth,nonnegative) Husimi function

Hsψ(x) = N |〈ψx,s, ψ〉|2 , x ∈ T2 . (3.8)

Applying definition (3.2) to the present framework, a sequence of states (ψN ∈ HN)N→∞converges to the measure µ on T2 iff, for any given s > 0, the Husimi measures (H s

ψN)N→∞

weak-∗ converge to the measure µ.Following Schubert’s work [39], one can also consider anti-Wick quantizations (and

dual Husimi measures) in which the squeezing parameter s in integral (3.7) depends on thephase space point. Adapting the proofs of [39] to the torus setting, one shows that all thesequantizations are equivalent to one another.

Proposition 5. Choose two functions s1, s2 ∈ C∞(T2, (0,∞)). Then the two associatedanti-Wick quantizations become close to one another when N → ∞:

∀f ∈ C∞(T2), ‖OpAW,s1N (f )− OpAW,s2

N (f )‖ = O(N−1) .


3.2.2. Standard quantization of the baker’s map. Strictly speaking, the quantization of theclosed baker’s map Br is well defined only if the coefficients r are such that

N ri = Ni ∈ N, i = 0, 1, 2 . (3.9)

Yet, in the semiclassical limit N → ∞ one can, if necessary, slightly modify the ri byamounts � 1/N in order to satisfy this condition: such a modification is irrelevant for theclassical dynamics. Assuming (3.9), the quantization of Br on HN is given by the followingunitary matrix on the position basis [2]:

Br,N = F−1N

FN0

FN1

FN2

. (3.10)

Here FNi denotes the Ni-dimensional discrete Fourier transform

(FNi )jk = Ni−1/2 e−2iπjk/Ni , j, k = 0, . . . Ni − 1. (3.11)

Since we already have a quantization for Br and the hole is a rectangle H = R1 ={r0 � q < 1 − r2}, a natural choice to quantize Br is to project on the positions q ∈[0, 1) \ [r0, 1 − r2) and then apply Br,N . One gets the following open propagator on theposition basis [35]:

Br,N = F−1N

FN0

0

FN2

. (3.12)

This is the ‘standard’ quantization of the open baker’s map Br. These matrices obviouslycontract on HN . The semiclassical connection between the matrices Br,N (respectively, Br,N )and the classical map map Br (respectively,Br) has been analysed in detail in [31]; this analysisimplies property (3.4) with respect to the Weyl quantization. Below we give an alternativeproof of property (3.4) for the open propagatorBr,N , with respect to the anti-Wick quantization.The proof uses the semiclassical propagation of coherent states analysed in [8].

To start with, we precisely give the continuity set of B−1r :

C(B−1r ) = R0 � R2 ,

where R0 = {q ∈ (0, 1), p ∈ (0, r0)} ⊂ Br(R0) and similarly for R2: in the case of thesymmetric baker, these are the open grey rectangles in the right plot of figure 1. On theother hand, the hole H−1 = Br(R1) = {q ∈ [0, 1), p ∈ [r0, 1 − r2)} (this is the white strip,including the vertical side). The discontinuity set D(B−1

rsym) is shown in figure 4 (pink/light

grey lines in the left plot).

Any observable a ∈ C∞c (C(B

−1r ) � H

−1) is supported at a distance � δ > 0 from the

discontinuity set D(B−1r ). Let us select some smooth s ∈ C∞(T2, (0,∞)) and consider the

corresponding anti-Wick quantization (see (3.7)). From definition (3.7), the operator OpAW,s

only involves coherent statesψx,s(x) located at a distance � δ > 0 fromD(B−1r ). Adapting the

proof of [8, proposition 5] to the general open bakerBr, one can show the following propagationfor such states:

B†r,N ψx,s(x) = 1M−1(x) eiθ(x,N) ψx ′,s ′(x ′) + OHN

(e−N C(s,δ)) , N → ∞ . (3.13)

Here x ′ = B−1r (x), s ′(x ′) = s(x)/r2

ε if x ′ lies inside the rectangle Rε , and θ(x,N) is a phasewhich can be explicitly computed.

Through the symplectic change in variable y = B−1r (x) for y ∈ M , one gets

B†r,N OpAW,s(a) Br,N = OpAW,s ′1M × (a ◦ Br)) + OB(HN )(e

−N C ′(s,δ)) .


Figure 4. On the left, we show the backward trapped set �+ for the symmetric open baker Brsym

(black) and the discontinuity set D(B−1rsym ) (pink/light grey). Their union gives �+ �D−∞, which

contains the support of semiclassical measures (see theorem 1, (i)). The dotted lines indicateidentical points on T

2. On the right, we show the backward image B−1rsym (D(B

−1rsym )) involved in

theorem 1, (iii) (pink/light grey), and the projection on T2 of the set � \�′′ defined in (2.10)

(red/dark grey).

The function s ′ is obtained by taking s ′(y) = s(Br(y))/r2ε for y ∈ B−1

r (supp a ∩ Rε) andsmoothly extending the function to s ′ ∈ C∞(T2, (0,∞)). Since the quantizations withparameters s and s ′ are equivalent (proposition 5), we have proven property (3.4) for thefamily (Br,N ). �

4. Fractal Weyl law for the quantized open baker

In this section, which mainly presents numerical results, we exclusively consider the openbaker’s maps Br and their quantizations (3.12). Our aim is to investigate the precise notion ofdimension entering the fractal Weyl law conjectured below through a numerical study whichcomplements the one performed in [30]. Still, we believe that most statements should hold aswell if we replace Br by a map T obtained by restricting an Anosov diffeomorphism T outsidesome ‘small hole’, as described in [6].

From the explicit formula (3.12), the subspace of HN spanned by {qj , N0 � j < N−N2}is in the kernel of Br,N . We call the spectrum of Br,N on the complementary subspace‘nontrivial’. This spectrum is situated inside the unit disc. In the semiclassical limitN → ∞,most of it accumulates near the origin, which corresponds to ‘short-living’ eigenvalues [38].We rather focus on ‘long-living’ eigenvalues, situated in some annulus away from the origin.By analogy with the case of potential scattering (1.1), a fractal Weyl law for the semiclassicaldensity of ‘long-living’ eigenvalues was conjectured in [31].

Conjecture 1 (fractal Weyl law). Let Br,N be the quantized open baker’s map described insection 3.2. Then, for any radius 0 < r < 1, there exists Cr � 0 such that

n(N, r)def= #{λ ∈ Spec(Br,N ), : |λ| � r} = Cr N

d + o(Nd), N → ∞ . (4.1)

The eigenvalues are counted with multiplicities, and 2d is an appropriate fractal dimension ofthe trapped set K .


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1dim

0

0.1

0.2

0.3

σ

Baker 1/32-2/3Fitting the Weyl law

d dI 0

Figure 5. Standard deviations when fitting the Weyl law (4.1) to various dimensions, integratedon 0.1 � r � 1. The two marks on the horizontal axis indicate the theoretical values dI and d0.

4.1. Which dimension plays a role?

In the proofs for upper bounds of the Weyl law (1.1), the exponent d is defined in terms of theupper Minkowski dimension of the trapped setK [15,41,42,49]. In the case of the open bakerBr, we therefore expect that the exponent d appearing in the conjecture (respectively, d + 1) isgiven by the Minkowski dimension of the Cantor set Cr (respectively, �+), which is equal toits box, Hausdorff and packing dimensions. We call this theoretical value d0.

For the symmetric baker Brsym , the Hausdorff dimension dH (�+) = d0 + 1 happens to beequal to the Hausdorff dimension of the natural measure µnat, defined by

dH (µnat) = infA⊂T2, µnat(A)=1

dH (A) .

For a nonsymmetric baker’s mapBr, these two Hausdorff dimensions only satisfy the inequalitydH (µnat) � dH (�+) (see the explicit expressions below). We want to investigate a possible‘role’ of the natural eigenmeasure µnat regarding the structure of the quantum spectrum. It istherefore legitimate to ask the following question.

Question 1. Is the correct exponent in the Weyl law (4.1) given by dI = dH (µnat)− 1 insteadof d0 = dH (�+)− 1?

Here the suffix I indicates that dI is sometimes called the information dimension [16]. Asmentioned above, both dimensions are equal for the symmetric bakerBrsym , for which the Weyllaw (1) has been numerically tested in [30]. They are also equal in the case of a closed mapon T2: in this case, the Weyl law has exponent 1 (the whole spectrum lies on the unit circle),and we have dH (T2) = dH (µL) = 2.

For a nonsymmetric baker Br, the two dimensions take different values:

d0 is the solution of rd0 + rd2 = 1, while dI = r0 logp0 + r2 logp2

r0 log r0 + r2 log r2.

To answer the above question, we considered a very asymmetric baker, taking rasym withr0 = 1/32, r2 = 2/3. The two dimensions then take the values d0 ≈ 0.493, dI ≈ 0.337. Wecomputed the counting function n(N, r) for several radii 0.1 � r � 1 and several values ofN . We then tried to fit the Weyl law (4.1) with an exponent d varying in a certain range andcomputed the standard deviations (see figure 5). The numerical result is unambiguous: the


0 0.2 0.4 0.6 0.8 1r

0

2

4

6

8

10

n(N

,r)/

N^d

N=1728N=2304N=2880N=3456N=4032N=4608N=5184N=5760

Baker 1/32 - 2/3 Rescaled resonance counting

0 0.2 0.4 0.6 0.8 1r

0

200

400

n(N

,r)

N=1728N=2304N=2880N=3456N=4032N=4608N=5184N=5760

Baker 1/32 - 2/3Unscaled resonance counting

Figure 6. Top: spectral counting function for the asymmetric baker Brasym , for various valuesof Planck’s constant N . Bottom: same curves vertically rescaled by N−d0 . The thick tick markindicates the radius

√�nat corresponding to the natural measure.

best fit clearly occurs away from dI , but it is close to d0. This numerical test rules out thepossibility that dI provides the correct exponent of the Weyl law and suggests to indeed taked = d0.

To further illustrate the Weyl law for the asymmetric baker Brasym , we plot in figure 6(top) the counting functions n(N, r) as a function of r ∈ (0, 1), for several values of N . Onthe bottom plot, we rescale n(N, r) by the power N−d0 : the rescaled curves almost perfectlyoverlap, indicating that scaling (4.1) is correct.

Remark 2. In figure 6 (right) the rescaled counting function seems to converge to a functionwhich is strictly decreasing on an interval [λmin, λmax], where λmin ≈ 0.1, λmax ≈ 0.9. Thisimplies that the spectrum of Brasym,N becomes dense in the whole annulus {λmin � |λ| � λmax},when N → ∞. Therefore, at this heuristic level, for any λ ∈ [λmin, λmax] one may consider

sequences of eigenvalues (λN)N�1 with the property |λN | N→∞−→ λ. In particular, we mayconsider sequences converging to

√�nat.


5. Localization of resonant eigenstates

We now return to the general framework of an open map T on a subsetM of a compact phasespace M , which is quantized into a sequence of operators (TN)N→∞ according to axioms 2.We want to study the semiclassical measures associated with the long-living eigenstates of TN .

To start with, we fix some λm ∈ (0, 1), such that, for N large enough, SpecTN ∩{|λ| � λm} �= ∅. We can then consider sequences of eigenstates (ψN)N→∞ such that, forN large enough,

TN ψN = λN ψN , ‖ψN‖ = 1 , |λN | � λm . (5.1)

The role of the (quite arbitrary) lower boundλm > 0 is to ensure that the eigenstates we considerare ‘long-living’. Up to extracting a subsequence, we can assume that (ψN) converges to acertain semiclassical measure µ (see section 3.1.2).

To state our result, we first need to analyse the continuity sets of the backward iterates ofT . For any n � 1, the map T −n is defined on the set M−n. Its continuity set C(T −n) can beobtained iteratively through

C(T −n) = T (C(T −n+1) ∩ C(T )) , n � 2 .

The set M−n of points escaping exactly at time (−n) can also be split between its continuitysubset

CM−ndef= C(T −n+1) ∩ M−n = T n−1(C(T n−1) ∩ H

−1) ,

which is a union of open connected sets, and its discontinuity subset DM−n = M−n\CM−n,which has Minkowski content zero (in the case n = 1, we take CM−1 = H

−1). Using this

splitting of the M−n, decomposition (2.5) can be recast into

M = C−∞ �D−∞ � �+, where C−∞ =( ∞⊔n=1

CM−n

), D−∞ =

( ∞⊔n=1

DM−n

).

(5.2)

The set C−∞ consists of points which eventually fall in the hole when evolved through T −1

and remain at a finite distance from D(T −1) all along their transient trajectory.We can now state our main result concerning semiclassical measures.

Theorem 1. Assume that a sequence of eigenstates (ψN)N→∞ of the open quantum map TN ,with eigenvalues |λN | � λm > 0, converges to the semiclassical measure µ on M . Then thefollowing hold.

(i) The support of µ is a subset of �+ �D−∞.(ii) If µ(C(T −1)) > 0, there exists � ∈ [λ2

m, 1] such that the eigenvalues (λN)N→∞ satisfy

|λN |2 N→∞−→ � .

For any Borel subset S not intersecting D(T −1), one has µ(T −1(S)) = �µ(S).(iii) If µ(D(T −1)) = µ(T −1(D(T −1))) = 0, then µ is an eigenmeasure of T , with the decay

rate �.

Proof. To prove the first statement, let us choose some n � 1 and take an observablea ∈ C∞

c (CM−n). Every point x ∈ supp a has the property that for any 0 � j < n − 1,

T −j (x) ∈ C(T −1), while T −n+1(x) ∈ H−1

. Applying iteratively property (3.4), one finds thatin the semiclassical limit

(T†N)

n OpN(a) (TN)n ∼ 0 .


We take into account the fact that ψN is a right eigenstate of TN :

〈ψN, OpN(a)ψN 〉 = |λN |−2n 〈ψN, (T †N)

n OpN(a) (TN)n ψN 〉 .

Using |λN | � λm, these two expressions imply µ(a)def= ∫

a dµ = 0. Since n and a arearbitrary, this shows µ(C−∞) = 0, which is the first statement.

From the assumption in (ii), we may select a ∈ C∞c (C(T

−1)) such that∫a dµ > 0. The

first iterate a ◦ T is supported in C(T ) ⊂ M . Applying (3.4), we get

|λN |2 〈ψN, OpN(a)ψN 〉 ∼ 〈ψN, OpN(a ◦ T )ψN 〉 . (5.3)

For N large enough, the matrix element 〈ψN, OpN(a)ψN 〉 > µ(a)/2, so we may divide theabove equation by this element and obtain

|λN |2 N→∞−→ µ(a ◦ T )µ(a)

.

We call � � |λm|2 this limit, which is obviously independent of the choice of a. For anyBorel set S ⊂ C(T −1), one can approximate 1S by smooth functions supported in C(T −1) toprove that

µ(T −1(S)) = �µ(S) .

If S ⊂ H−1

, we know that µ(S) = 0 and T −1(S) = ∅, so the above equality still makes sense.This proves (ii).

To obtain (iii), we split any Borel set S into S = (S ∩D(T −1)) �CS. From (ii), we haveµ(T −1(CS)) = �µ(CS). By assumption, µ(S ∩D(T −1)) = µ(T −1(S ∩D(T −1))) = 0, sowe get µ(T −1(S)) = �µ(S). �

5.1. Semiclassical measures of the open baker’s map

In this section we apply the above theorem to the case of some open baker’s mapBr, quantized asin section 3.2.2. We also numerically compute the Husimi measuresH 1

ψNassociated with some

quantum eigenstates (we choose the isotropic squeezing s = 1 for convenience): although wecannot really go to the semiclassical limit, we hope that forN ∼ 1000 these Husimi measuresalready give some idea of the semiclassical measures.

5.1.1. Applying theorem 1. To simplify the presentation we will restrict the discussion to thesymmetric baker Brsym , which will be denoted by B in short. The discontinuity set D(B−1),its backward image and the trapped sets were described in section 2.2.2 and section 3.2.2and plotted in figure 4. The sets M−n and their continuity subsets are also simple to describe

using symbolic dynamics. For any n = 1, we know that CM−1 = H−1

is the open rectangle{q ∈ [0, 1), p ∈ (1/3, 2/3)}. For n � 2, CM−n is the union of 2n−1 horizontal rectangles,indexed by the n-sequences ε = ε−1 . . . ε−n such that ε−i ∈ {0, 2}, i = 1, . . . , n − 1, whileε−n = 1. Each such rectangle is of the form

{q ∈ (0, 1), p ∈ (p(ε), p(ε) + 3−n)

}, where

p(ε) ≡ · ε−1 . . . εn in ternary decomposition. Some of those rectangles are shown in figure 2(centre). The union of all these rectangles (for n � 1) makes up C−∞. Its complement�+ �D−∞ is given by

�+ �D−∞ = ([0, 1)× Crsym ) ∪D(B−1) .

This set is shown in figure 4 (left).In figure 7, we plot the Husimi densities H 1

ψNof some (right) eigenstates of Br,N , for the

symmetric baker and an asymmetric one, r2 = (1/9, 5/9, 1/3).


Figure 7. Husimi densities of (right) eigenstates of Br,N (black=large values, white=0). Top: 3eigenstates of Brsym ,N with |λN | ≈ √

�nat . Bottom left, centre: two eigenstates of Brsym ,N withdifferent |λN |. Bottom right: one eigenstate of Br2,N .

Remark 3. We note that all Husimi functions are indeed very small in the horizontal rectanglesM−n for n = 1, 2 (in the case N � 1500) and also n = 3 for N = 4200. Using (3.13), onecan refine the proof of theorem 1 to show that H 1

ψN(x) = O(e−cN ) for x ∈ C−∞, N → ∞.

Remark 4. Although this is not proven in our theorem, all the Husimi functions that we havecomputed are very small on the set D(B−1)\�+ ⊂ {q = 0} (see the left plot in figure 4). Onthe other hand, some of these Husimi functions are large on D(B−1) ∩ �+, so we cannot ruleout the possibility that some semiclassical measures µ charge this set. For instance, in someof the Husimi plots (e.g. the bottom left in figure 7), we clearly see a strong peak at the originand lower peaks on other points of D(B−1) ∩ �+. We call the components of an eigenstatelocalized on D−∞ ‘diffractive’.

From these observations, we state the following conjecture.

Conjecture 2. Let Br be an open baker’s map, quantized by section 3.2.2. Then

• all long-living semiclassical measures are supported on �+,• ‘almost all’ long-living eigenstates are nondiffractive, that is, their weights onD(B−1) andB−1(D(B−1)) are negligible in the semiclassical limit. The corresponding semiclassicalmeasures are then eigenmeasures of Br.

In the next section we further comment on the above Husimi plots.

5.2. Abundance of semiclassical measures

Let us draw some consequences from theorem 1 and the following remarks. Statement (ii) ofthe theorem strongly constrains the converging sequences of eigenstates: a sequence (ψN) can


converge to some measure µ (with µ(C(B−1)) > 0) only if the corresponding eigenvalues(λN) asymptotically approach the circle of radius

√�.

We take for granted the density argument in remark 2 and assume that a ‘dense interval’[λmin, λmax] ⊂ (0, 1] exists for any open baker’s mapBr. Hence, for any� ∈ [λ2

min, λ2max] there

exist many sequences of eigenstates of Br,N , such that |λN |2 N→∞−→ �. From our conjecture 2,almost any semiclassical measure associated with such a sequence will be an eigenmeasurewith the decay rate �. Therefore, two converging sequences (ψN), (ψ ′

N) associated withlimiting decays � �= �′ will necessarily converge to different eigenmeasures. This alreadyshows that the semiclassical eigenmeasures generated by all possible sequences in the annulus{λmin � |λ| � λmax} form an uncountable family.

According to section 2.3.2, for each decay rate � ∈ (0, 1), there exist uncountablymany eigenmeasures. A natural question thus concerns the variety of semiclassical measuresassociated with a given � ∈ [λ2

min, λ2max].

Question 2.For a given � ∈ [λ2

min, λ2max], what are the semiclassical measures of Br of the decay

rate �?

• Is there a unique such measure?• Otherwise, is some limit measure ‘favoured’, in the sense that ‘almost all’ sequences (ψN)

with |λN |2 → � converge to µ?• Can the natural measure µnat be obtained as a semiclassical measure?

The same type of questions were asked by Keating and coworkers in [20]. At present weare unable to answer them rigorously for the quantum open baker Br,N .

From a heuristic point of view we notice the following features in the plots of figure 7. Thethree top plots correspond to eigenvalues |λN | close to the value

√�nat ≈ 0.8165. However,

the Husimi measures on the left and centre seem very different from the natural measure (thelatter is approximated by the black rectangles in figure 4). These two Husimi functions seemto charge different parts of �+. Only the rightmost state seems compatible with a convergenceto µnat.

As commented in remark 4, the bottom-left state has strong concentrations onD(B−1)∩�+.For the various values ofN that we have investigated, this concentration seems characteristic ofthe eigenstate with the largest |λN |. The discontinuities of Br manage to ‘trap’ these quantumstates better than periodic orbits on C(B−1) ∩ �+. Such states seem ‘very diffractive’.

Finally, the bottom-centre state, with a smaller eigenvalue, clearly shows a self-similarstructure along the horizontal direction, with the probability inside the hole being higher thanfor the top eigenstates. This feature had already been noticed in [20] for averages over Husimifunctions with comparable |λN |.

In section 6 we will address question 2 for a different quantization of the symmetric openbaker, namely, the Walsh-quantized open baker, where one can compute some semiclassicalmeasures explicitly.

Before that, we explain how to construct approximate eigenstates (pseudomodes) for thequantum baker Brsym,N .

5.3. Pseudomodes and pseudospectrum

From the explicit representation of eigenmeasures given in proposition 2, it is possible toconstruct approximate eigenstates of TN by backward propagating wavepackets localized onthe set �(1)+ = �+ ∩H .


We call these approximate eigenstates pseudomodes, by analogy with the recent literatureon nonself-adjoint semiclassical operators (see, e.g. [3, 10]). These papers deal withpseudodifferential operators obtained by quantizing complex-valued observables, and theyconstruct pseudo-modes of error O(h∞), microlocalized at a single phase space point.

Our pseudomodes will be less precise and will be microlocalized on a countable set. Wewill restrict ourselves to the case of the open symmetric baker B = Brsym , but our constructionworks for any Br and can probably be extended to other maps or phase spaces.

Inspired by the pointwise eigenmeasures (2.15), we construct approximate eigenstates bybackward evolving coherent states localized in �(1)+ . Precisely, for any x0 ∈ �

(1)+ , s > 0 and

λ ∈ C, |λ| < 1, N ∈ N, we define the following quantum state:

�sλ,x0

def=√

1 − |λ|2∑n�0

λn B†nN ψx0,s , (5.4)

where ψx0,s is the coherent state defined in section 3.2.1 and BN is the standardquantization of B.

Proposition 6. Consider the symmetric open baker B = Brsym and its quantization (BN). Fixλ ∈ C with |λ| < 1.

(i) Choose ε > 0 small and call α = (1 − ε)(log 1/|λ|/log 3). For any N ∈ N∗, one canchoose a point x0(N) ∈ �

(1)+ and a squeezing parameter s = s(N) > 0 such that the

states(�N

def= �s(N)

λ,x0(N)

)N→∞

defined in (5.4) satisfy

‖�N‖ = 1 + O(N−α), ‖(BN − λ)�N‖ = O(N−α) , N → ∞.

(ii) For any x0 ∈ �(1)+ , one can select the points (x0(N)) such that x0(N)

N→∞−→ x0. In thiscase, the sequence (�N)N→∞ converges to the eigenmeasure µx0,� described in (2.15),with � = |λ|2.

This proposition shows that, for any α > 0 andN large enough, theN−α-pseudospectrumofBN contains the disc {|λ| � 3−α(1+ε)}. We note that the errors are not very small and increasewhen |λ| → 1. We should emphasize that, in our numerical trials, we never found eigenstatesof BN with Husimi measures looking like µx0,�.

Proof of the proposition. To control series (5.4), we would like to ensure that the evolvedcoherent state B†j

N ψx0(N),s(N) remains close to an approximate coherent state ψx−j ,s−j (as in(3.13)) up to large times j . For this we need the points x−j = B−j (x0(N)) to stay ‘far’ fromthe discontinuity set D(B−1), and we also need all ψx−j ,s−j to be microlocalized in a smallneighbourhood of x−j . Due to the hyperbolicity of B−1, the second condition constrains thetimes j to be smaller than the Ehrenfest time [8]

nE = (1 − ε)logN

log 3. (5.5)

Here ε > 0 is the small parameter in the statement of the proposition.To identify a good ‘starting point’ x0(N), we set n = [nE] and consider the following

subset of T2, for δ, γ > 0:

Dn,δ,γdef= {(q, p) ∈ T2, q ∈ (1/3 + δ, 2/3 − δ), ∀k ∈ Z, |p − k

3n| > γ } ∩ �(1)+ .

We first show that this set is nonempty if δ < 1/9 and γ = 3−nγ ′, γ ′ < 1/9. Takeq ∈ (1/3 + δ, 2/3 − δ) arbitrary and select p = p(ε) with the following properties: takeall indices ε−j ∈ {0, 2}, j � 1, so that p ∈ Crsym , and require furthermore that the wordε−n−1ε−n−2 ∈ {02, 20}. The point (q, p) is then in Dn,δ,γ .


Let us take any point x0(N) in that set. It automatically lies in C(B−n), so its backwarditerates stay away fromD(B−1) at least until the time n. Let us select the squeezing parameters(N) = s0 = N−1+ε. A simple adaptation of [8, proposition 5] implies the following estimate:

∃c, C > 0, ∀j, 0 � j � nE, ‖B†jN ψx0,s0 − eiθj ψx−j ,s−j ‖ � C e−c Nε

. (5.6)

Here we can take c = min(δ2, γ ′2) and C is uniform w.r.t. the initial point x0. From thevalues of x0, s0, one checks that the components 〈qk, ψx0,s0〉 for k/N �∈ [1/3, 2/3] are of orderO(e−cNε

): this state is (very) localized in the hole H = R1, so

BN ψx0,s0 = O(e−c Nε

) .

Similarly, for any j � nE , the state ψx−j ,s−j is localized inside a certain connected componentof Mj+1 (one of the pink/grey rectangles in figure 2, left). In particular, the components of〈qk, ψx−j+1,s−j+1〉 are exponentially small in H . Therefore,

∀j, 0 � j � nE, BN B†Nψx−j ,s−j = ψx−j ,s−j + O(e−c Nε

) .

Summing all terms j � nE in (5.4) and estimating the remaining series using ‖BN‖ � 1, weobtain

‖(BN − λ)�sλ,x0

‖ = O(|λ|nE ) .From the definition of nE , we have |λ|nE = N−α .

The asymptotic normalization of �sλ,x0

is proven by estimating the overlaps betweencoherent states ψx−j ,s−j , ψx−j ′ ,s−j ′ for j, j ′ � nE . Because the setsMj+1 andMj ′+1 are disjointfor j �= j ′, the above mentioned localization properties imply that

∀j, j ′ � nE, 〈ψx−j ,s−j , ψx−j ′ ,s−j ′ 〉 = δj ′,j + O(e−c Nε

),

and the normalization estimate follows. This achieves the proof of (i).To prove (ii), let us consider an arbitrary x0 = (q0, p0) ∈ �

(1)+ . If q0 �∈ {1/3, 2/3}, we

take δ small enough such that q0 ∈ (1/3 + δ, 2/3 − δ) and set q0(N) = q0. Otherwise,we may let δ slowly decrease with N and find a sequence (q0(N)) such that q0(N) ∈(1/3 + δ(N), 2/3 − δ(N)) and q0(N)

N→∞−→ q0. On the other hand, p0 ∈ Crsym = p(ε) fora certain sequence ε, with all ε−j ∈ {0, 2}. For each N , we inspect the word ε−n−1ε−n−2

of that sequence, where n = [nE] (see (5.5)). If this word is in the set {02, 20} we keepp0(N) = p0, otherwise we replace this word by 02 (keeping the other symbols unchanged) to

define p0(N). The point x0(N) = (q0(N), p0(N)) is then in Dn,δ,γ , and x0(N)N→∞−→ x0.

The convergence to the measure µx0,� is due to the localization of the coherent statesψx−j ,s−j for j � nE . �

6. A solvable toy model: Walsh-quantization of the open baker

In this section we study an alternative quantization of the open symmetric baker Brsym ,

introduced in [30, 31]. A similar quantization of the closed baker Brsym was proposed andstudied in [11, 36, 45], mainly motivated by research in quantum computation. From now on,we will drop the index rsym from our notations and call B = Brsym .

A ‘simplified’ quantization of B was introduced in [30, 31], as a N × N matrix BN ,obtained by only keeping the ‘skeleton’ of BN (see (3.12)). Although BN can be defined forany N , its spectrum can be explicitly computed only if N = 3k for some k ∈ N. As explainedin those references, BN can be interpreted as the ‘Weyl’ quantization of a multivalued map Bbuilt upon B. In the present work we will stick to a different interpretation of BN , valid in


the case N = 3k: one can then introduce a (Walsh) quantization for observables on T2, whichis not equivalent to the anti-Wick quantization of section 3.2.1. We will check below thatthe matrices BN satisfy property (3.4) with respect to that quantization and the open baker B.Finally, this Walsh quantization is also suited to quantizing observables on the symbol space�: the matrices BN then quantize the open shift σ . We will see that this interpretation is more‘convenient’, because it avoids problems due to discontinuities.

We first recall the definition of the Walsh-quantization of observables on T2 (or �) andthe associated Walsh–Husimi measures.

6.1. Walsh transform and coherent states

6.1.1. Walsh coherent states. The Walsh-quantization of observables on T2 uses thedecomposition of the quantum Hilbert space HN into a tensor product of ‘qubits’, namely,HN = (C3)⊗k (clearly, this makes sense only for N = 3k). Each discrete position qj = j/N

can be represented by its ternary sequence qj = 0 · ε0ε2 · · · εk−1, with symbols εi ∈ {0, 1, 2}.Accordingly, each position eigenstate qj can be represented as a tensor product:

qj = eε0 ⊗ eε2 ⊗ · · · ⊗ eεk−1 = |[ε0ε2 . . . εk−1]k〉 ,where {e0, e1, e2} is the canonical (orthonormal) basis of C3. The notation on the right-handside emphasizes the fact that this state is associated with the cylinder [ε] = [ε]k with k symbolson the right of the comma and no symbols on the left. Its image on T2 is a rectangle [ε]k ofheight unity and width 3−k = 1/N .

The Walsh-quantization consists of replacing the discrete Fourier transform (3.11) onHN by the Walsh(–Fourier) transform WN , which is a unitary operator preserving the tensorproduct structure of HN . We define it through its inverse W ∗

N , which maps the position basisto the orthonormal basis of ‘Walsh momentum states’: for any j ≡ ε0 . . . εk−1,

pj = W ∗N qj

def= F ∗3 eεk−1 ⊗ F ∗

3 eεk−2 ⊗ · · · ⊗ F ∗3 eε1 ⊗ F ∗

3 eε0

(here F ∗3 is the inverse Fourier transform on C3). To agree with our notations of section 2.2,

we will index the symbols relative to the momentum coordinate by negative integers, so theWalsh momentum states will be denoted by

pj = |[ε−k . . . ε−1]0〉 , where pj = j/N = 0 · ε−1 . . . ε−k .

This momentum state is associated with a rectangle of height 3−k and width unity.More generally, for any � ∈ {0, . . . , k} and any sequence ε = ε−k+� . . . ε−1 · ε0 . . . ε�−1,

one can construct a (Walsh-)coherent state

|[ε]�〉 def= eε0 ⊗ · · · eε�−1 ⊗ F ∗3 eε−k+� ⊗ · · ·F ∗

3 eε−1 . (6.1)

This state is localized in the rectangle [ε]� with height 3−k+� and width 3−�, so it still has anarea 1/N (all such rectangles are minimal-uncertainty or ‘quantum’ rectangles).

Like the squeezing parameter s of Gaussian wavepackets (see section 3.2.1), the index� describes the aspect ratio of the coherent state. When going to the semiclassical limitk → ∞, we will always select �(k) ∼ k/2, which corresponds to an ‘isotropic’ squeezing.One important difference between the Gaussian and the Walsh coherent states lies in the factthat the latter are strictly localized both in momentum and position. Another difference isthat, for each �, the �-coherent states make up a finite orthonormal basis of HN , instead of acontinuous overcomplete family.


6.1.2. Walsh-quantization for observables on �. As explained in [1], it is more natural toWalsh-quantize observables on the symbol space � than on T2. Indeed, if one equips � withthe metric structure

d�(ε, ε′) = max(3−n+ , 3−n−), n+ = min

{n � 0, εn �= ε′

n

},

n− = min{n � 0, ε−n−1 �= ε′

−n−1

},

the closed shift σ then acts as a Lipschitz map on�, and the hole {ε0 = 1} is at a finite distancefrom its complement in �. Indeed, the ‘lift’ from T2 to � has the effect to ‘blow up’ the linesof discontinuity of B.

The conjugacy J : � → T2 is also Lipschitz, so any Lipschitz function F ∈ Lip(T2)

is pushed to a function f = F ◦ J ∈ Lip(�). However, the converse is not true: if weuse the inverse map (J|�′)−1 : T2 → �′ and take an arbitrary f ∈ Lip(�), the functionF = f ◦ (J|�′)−1 is generally discontinuous on T2.

Let us select some � ∼ k/2. The Walsh-quantization of a function f ∈ Lip(�) is definedas the following operator on HN :

Op�N (f )def= 3k

∑[ε]�

|[ε]�〉〈[ε]�|∫

[ε]�

f dµL. (6.2)

Here the sum goes over all ‘quantum’ cylinders [ε]�, that is, over all 3k sequences ε =ε−k+� . . . ε−1 ·ε0 . . . ε�−1. The integral over f is performed using the uniform Bernoulli measureµL, which is equivalent to the Liouville measure on T2 (see the end of section 2.2). Note theformal similarity of this quantization with the anti-Wick quantization (3.7).

It is shown in [1, proposition 3.1] that this quantization satisfies axioms 1 (with Lipschitzobservables) and that two quantizations Op�1

N , Op�2N are semiclassically equivalent if both

�1, �2 ∼ k/2.By duality, we define the Walsh–Husimi measure of a quantum state ψN . The

corresponding density is constant in each �-cylinder [ε]�:

∀α ∈ [ε]� , WH�ψN(α) = 3k |〈[ε]�, ψN 〉|2 . (6.3)

This density is originally defined on � but can be pushed forward to T2. Semiclassicalmeasures are defined as the weak-∗ limits of sequences (WH�

ψN), where N = 3k → ∞ and

� = �(k) ∼ k/2. One first obtains a measure µ� on �, which can be pushed on T2 intoµ = J ∗µ� (equivalently, µ is the weak-∗ limit of the Walsh–Husimi measures on T2).

6.2. Walsh-quantization of the open baker

We now recall the Walsh-quantization of the open baker B = Brsym , as defined in [30, 31].Mimicking the standard quantization (3.12), we replace the Fourier transforms F ∗

N , FN/3 bytheir Walsh analogues WN , WN/3 (with N = 3k , k � 0), so that the Walsh-quantized openbaker is given by the following matrix on the position basis:

BNdef= W ∗

N

WN/3

0

WN/3

. (6.4)

For any set of vectors v0, . . . , vk−1 ∈ C3, this operator acts as follows on any tensor productstate v0 ⊗ · · · ⊗ vk−1:

BN(v0 ⊗ v1 . . .⊗ vk−1) = v1 ⊗ . . .⊗ vk−1 ⊗ F ∗3 v0 . (6.5)

Here F ∗3 = F ∗

3 π02, where π02 is the orthogonal projector on Ce0 ⊕ Ce2 in C3.


One can generalize the quantum-correspondence of [1, proposition 3.2] to the open shiftσ and prove the Walsh version of axioms 2.

Proposition 7.

(i) Take any f ∈ Lip(�). Then, in the limit N = 3k → ∞, � ∼ k/2,

B†N Op�N (f ) BN ∼ Op�N1{ε0 �=1} × f ◦ σ) .

Note that the function 1{ε0 �=1} × f ◦ σ ∈ Lip(�).

(ii) If we take F ∈ Lipc(C(B−1) � H

−1) and f = F ◦ J , we have

1{ε0 �=1} × f ◦ σ = 1M × F ◦ B) ◦ J .The points (i) and (ii) show that the family (BN) satisfies axioms 2 with respect to the map

B on T2 and quantization (6.2) of observables in Lip(T2). Point (i) alone shows that (BN) isa quantization of the open shift σ on �. As noted above, the latter interpretation allows us toget rid of problems of discontinuities.

Proof. Applying (6.5) to a coherent state |[ε]�〉, we get the exact evolution

B†N |[ε]�〉 = (1 − δε−1,1) |[σ−1(ε)]�+1〉 .

That is, the coherent state |[ε]�〉 is either killed if σ−1([ε]�) = ∞ or transformed into a coherentstate associated with the cylinder [σ−1(ε)]�+1 = σ−1([ε]�). This exact expression, which isthe quantum counterpart of the classical shift (2.9), should be compared with the approximateexpression (3.13). From there, a straighforward computation shows that, for any f ∈ Lip(�):

B†N Op�N (f ) BN = Op�+1

N ((1 − δε0,1)× f ◦ σ) .The semiclassical equivalence Op�+1

N ∼ Op�N finishes the proof of (i).

To prove (ii), we note that, if F ∈ Lipc(C(B−1)� H

−1) and f = F ◦J , the function F ◦B

is supported away fromD(B), and both functions 1{ε0 �=1} × f ◦ σ and 1{ε0 �=1} × F ◦B ◦ J aresupported inside�′′. Semiconjugacy (2.11) shows that these two functions are equal. Finally,we note that, for ε ∈ �′′, one has 1{ε0 �=1}(ε) = 1M ◦ J (ε). �

6.2.1. Spectrum of the Walsh open baker. The simple expression (6.5) allows us to explicitlycompute the spectrum of BN (see [31, proposition 5.5]). This spectrum is determined bythe two nontrivial eigenvalues λ− and λ+ of the matrix F ∗

3 . These eigenvalues have moduli|λ+| ≈ 0.8443, |λ−| ≈ 0.6838. The spectrum of BN has a gap: the long-living eigenvaluesare contained in the annulus {|λ−| � |λ| � |λ+|}, while the rest of the spectrum lies at theorigin. Most of the eigenvalues are degenerate. If we count multiplicities, the long-living (≡nontrivial) spectrum satisfies the following asymptotics when k → ∞:

∀r > 0, #{λj ∈ Spec(BN) , |λj | � r} = Cr 2k + o(2k) ,

Cr ={

1 , r < r0,

0 , r > r0,, r0

def= |λ−λ+|1/2 = 3−1/4 .(6.6)

The nontrivial spectrum is spanned by a subspace HN,long of dimension 2k . Since the trapped setforB = Brsym has dimension 2d = 2 (log 2/log 3), the above asymptotics agrees with the fractalWeyl law (4.1). Although the density of resonances (counted with multiplicities) is peakednear the circle {|λ| = r0}, the spectrum (as a set) densely fills the annulus {|λ−| � |λ| � |λ+|}when k → ∞ (see figure 8). In the next section we construct some long-living eigenstatesof BN and analyse their Walsh–Husimi measures. Due to the spectral degeneracies, there


-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-3 -2 -1 0 1 2 3

k=10k=15

Figure 8. Nontrivial spectrum of the Walsh open baker BN forN = 310 (circles) and 315 (crosses),using a logarithmic representation (horizontal = arg λj , vertical = log |λj |). We plot horizontallines at the extremal radii |λ±| of the spectrum (dashed), at the radius r0 = |λ−λ+|1/2 of highestdegeneracies (dotted) and at the radius corresponding to the natural measure (full).

is generally large freedom for selecting eigenstates (ψN) associated with a sequence ofeigenvalues (λN). Intuitively, this freedom should provide more possibilities for semiclassicalmeasures.

We mention that Keating and coworkers have recently studied the eigenstates of a slightlydifferent version of the Walsh–baker, namely, a matrix B ′

N obtained by replacing F3 by the‘half-integer Fourier transform’ (G3)jj ′ = 3−1/2e−2iπ(j+1/2)(j ′+1/2)/3 (see [19]).

6.3. Long-living eigenstates of the Walsh open baker

We first provide the analogue of theorem 1 for the Walsh–baker. We recall that a semiclassicalmeasureµ� (or its push-forwardµ) is now a weak-∗ limit of some sequence of Walsh–Husimimeasures (WH�

ψN)N=3k→∞, where ψN are eigenstates of BN and the index � ≈ k/2.

From the quantum-classical correspondence of proposition 7 and using proposition 4, wededuce the following corollary.

Corollary 1. Let µ� be a semiclassical measure for a sequence of long-living eigenstates(ψN, λN) of the Walsh–baker BN . Then

(i) µ� is an eigenmeasure for the open shift σ on � and the corresponding decay rate �satisfies � = limN→∞ |λN |2,

(ii) if µ�(�\�′′) = 0 (where �′′ is defined in (2.10)), then µ = J ∗µ� is an eigenmeasure ofthe open baker B, with the decay rate �.

From the structure of Spec(BN) explained above, there exist sequences of eigenvalues(λN)N→∞ converging to any circle of radius λ ∈ [|λ−|, |λ+|]. We also know that anysemiclassical measure is an eigenmeasure of σ , so it is meaningful to ask question 2 inthe present framework (setting λmax /min = |λ±|). We add the following question: are theresemiclassical measuresµ� such thatµ�(�′′) < 1? In the next section we give partial answersto these questions.


Concerning the last point in question 2, we note that the ‘physical’ decay rate forB = Brsym

is �nat = 2/3. The circle{|λ| = √

�nat}

is contained inside the annulus {|λ−| � |λ| � |λ+|}where the nontrivial spectrum of BN is semiclassically dense, although it differs from the circle{|λ| = r0} where the spectral density is peaked, see figure 8. Still, there exist semiclassicalmeasures of BN with eigenvalue�nat, and it is relevant to ask whetherµnat can be one of these.At present we are not able to answer that question. The next section shows that there are plentyof semiclassical measures with eigenvalue�nat, so even if µnat is a semiclassical measure with� = �nat, it is certainly not the only one.

6.4. Constructing the eigenstates of BN

In this section we construct one particular (right) eigenbasis of BN restricted to the subspaceHN,long of long-living eigenstates. The construction starts from the (right) eigenvectorsv± ∈ C3 of F ∗

3 associated with λ±. Note that these two vectors (which we take normalized)are not orthogonal to each other. For any sequence η = η0 . . . ηk−1, ηi ∈ {±}, we form thetensor product state

|η〉 def= vη0 ⊗ vη1 . . .⊗ vηk−1 .

Action (6.5) of BN implies that

BN |η〉 = λη0 |τ(η)〉 ,where τ acts as a cyclic shift on the sequence: τ(η0 . . . ηk−1) = η1 . . . ηk−1η0. The orbit{τ j (η), j ∈ Z} contains �η elements, where the period �η of the sequence η necessarilydivides k. The states {|τ j (η)〉, j = 0, . . . , �η − 1} are not orthogonal to each other but forma linearly independent family, which generates the BN -invariant subspace Hη ⊂ HN,long. The

eigenvalues of BN restricted to Hη are of the form λη,r = e2iπr/�η (∏�η−1j=0 ληj )

1/�η (with indicesr = 1, . . . , �η), and the corresponding eigenstates read as

|ψη,r〉 = 1√Nη,r

�η−1∑j=0

cη,r,j |τ j (η)〉 , cη,r,j =j−1∏m=0

ληm

λη,r

, (6.7)

where Nη,r > 0 is the factor which normalizes |ψη,r〉. Up to a phase, this state is unchanged ifη is replaced by τ(η). In the following subsections we explicitly compute the Walsh–Husimimeasures of some of these eigenstates.

6.4.1. Extremal eigenstates. The simplest case is provided by the sequence η = + + · · · +,which has period 1, so that |ψ+,N 〉 = |η〉 = v⊗k

+ is the (unique) eigenstate associated with thelargest eigenvalue λ+ (this is the longest-living eigenstate). For any choice of index 0 � � � k,the Walsh–Husimi measure of |ψ+,N 〉 factorizes:

for all �-rectangle [ε]�, WH�ψ+,N

([ε]�) =�−1∏j=0

|〈v+, eεj 〉|2−k+�∏j=−1

|〈v+, F∗3 eεj 〉|2 .

The second product involves the vectorw+def= F3v+, with componentsw+,ε = (1−δε,1)v+,ε/λ+.

Following the notations of section 2.3.5, let µ�P+be the Bernoulli eigenmeasure of σ with

weights P+,ε = |v+,ε |2, P ∗+,ε = |w+,ε |2. The above expression shows that the Husimi measure

WH�ψ+,N

is equal to the measure µ�P+, conditioned on the grid formed by the �-cylinders. Since

the diameters of the cylinders decrease to zero as k → ∞, �(k) ∼ k/2, the Husimi measures(H�

ψ+,N) converge to µ�P+

.


Figure 9. Walsh–Husimi densities for the extremal eigenstates ψ+,N , ψ−,N of BN , with N = 36,� = 3. These are coarse-grained versions of the Bernoulli measures µP+ and µP− .

One can similarly show that the Husimi functions of the eigenstatesψ− = v⊗k− , associated

with the smallest nontrivial eigenvalue λ−, converge to the Bernoulli eigenmeasure µ�P− , withweights P−,ε = |v−,ε |2, P ∗

−,ε = |w−,ε |2, where w− = F3v−.In figure 9 we plot the Walsh–Husimi densities (pushed forward on T2) for ψ+,N and

ψ−,N , using the ‘isotropic” � = k/2. These give a clear idea of the self-similar structure ofthe respective semiclassical measures µP+ and µP− . The weights have the approximate valuesP+ ≈ (0.579, 0.287, 0.134), P− ≈ (0.088, 0.532, 0.380).

Considering the fact that the eigenvalues λN close to the circles of radii |λ+| and |λ−| havesmall degeneracies, we propose the following conjecture.

Conjecture 3. Any sequence of eigenstates (ψN)N→∞ with eigenvalues |λN | → |λ+|(respectively, |λN | → |λ−|) converges to the semiclassical measure µ�P+

(respectively, µ�P− ).

This conjecture can be proven for the version of the Walsh baker B ′N studied in [19]: in

this case the two eigenvectors of G∗3 replacing v± are orthogonal to each other, which greatly

simplifies the analysis. The limit measure µ�P′+

is then the ‘uniform’ measure on the trapped

set {εn �= 1, n ∈ Z}, with P′+ = P′

+∗ = (1/2, 0, 1/2).

6.4.2. Semiclassical measures in the ‘bulk’. In this section we investigate some eigenstatesof the form (6.7), with eigenvalues λN situated in the ‘bulk’ of the nontrivial spectrum, that is,|λN | ∈ (|λ−|, |λ+|).

In the next proposition we show that there can be two different semiclassical measureswith the same decay rate �. This answers in the negative the first point in question 2.

Proposition 8. Choose a rational number t = m/n, with n � 1, 1 � m � n− 1.

(i) Select a sequence ηn with m (+) and n − m (−). For all k′ � 1, form the repeatedsequence (ηn)

k′and choose r = r(k′) ∈ Z arbitrarily. Then the sequence of eigenstates

(ψ(ηn)k′,r(k′))k=nk′→∞ converges to the semiclassical measure µ�ηn , which is a linear


Figure 10. Walsh–Husimi densities of two eigenstates ψη,0 constructed from the sequencesη4 = + − +− (left) and η′

4 = + + −− (right).

combination of Bernoulli measures for the iterated shift σn (see (6.12)). This measure

is independent of the choice of (r(k′)) and has a decay rate �tdef= |λ1−t

− λt+|2. Its push-forward is an eigenmeasure of B.

(ii) If ηn and η′n are two sequences with m (+) and n − m (−), which are not related by a

cyclic permutation, then the semiclassical measures µ�ηn , µ�η′n

are mutually singular and

so are their push-forwards on T2.

Note that the radius√�t lies in the ‘bulk’ of the nontrivial spectrum, but can be different

from the radius r0 where the spectral density is peaked. In figure 10 we plot the Husimifunctions of two states ψη4,0, ψη′

4,0 constructed from two 4-sequences η4, η′4 not cyclically

related. The two functions, which give a rough idea of the limit measures µη4, µη′

4, seemingly

concentrate on different parts of the phase space.

Proof. In short we call η = (ηn)k′

which has a period �η = �ηn and r = r(k′). From (6.7),each state |ψη,r〉 is a combination of �η states |τ j (η)〉, and its eigenvalue has an exact modulus√�t . When k′ → ∞, these states are asymptotically orthogonal to each other. Indeed, their

overlaps can be decomposed as

〈η|τ j (η)〉 = (〈ηn|τ j (ηn)〉)k′

j = 0, . . . , �η − 1 ,

and for any j �= 0 mod �η we have ηn �= τ j (ηn), which implies |〈ηn|τ j (ηn)〉| � c, with

cdef= |〈v+, v−〉|2 < 1. As a result, the normalization factor of ψη,r satisfies

Nη,r =�η−1∑j=0

|cη,r,j |2 + O(ck′) , k′ → ∞ .

To study the semiclassical measures of the sequence (ψη,r )k′→∞, we fix some cylinder[α] = [α−l′ . . . α−1 · α0 . . . αl−1] and compute the weight of the measures H�

ψη,r. If k is


large enough and � ∼ k/2, the conditions � > l and k − � > l′ are fulfilled, and this weightcan be written as

H�ψη,r([α]) = 〈ψη,r |�[α]|ψη,r〉 =

�η−1∑j,j ′=0

cη,r,j cη,r,j ′ 〈τ j ′(η)|�[α]|τ j (η)〉 , (6.8)

where the projector on [α] is a tensor product operator:

�[α] = πα0 ⊗ πα1 . . . παl−1 ⊗ (I )⊗k−l−l′ ⊗ F ∗

3 πα−l′F3 ⊗ · · · ⊗ F ∗3 πα−1F3 .

The tensor factor (I )k−l−l′−1 implies that each matrix element 〈τ j ′

(η)|�[α]|τ j (η)〉 containsa factor (〈τ j ′

(ηn)|τ j (ηn)〉)k′−O(1); for the same reasons as above, this element is O(ck′) if

j �= j ′. We are then led to consider only the diagonal elements j = j ′:

∀j = 0, . . . , �η − 1, 〈τ j (η)|�[α]|τ j (η)〉 =l−1∏i=0

Pηj+i ,αi

l′∏i ′=1

P ∗ηj−i′ ,α−i′ . (6.9)

As above the weights P±,ε = |v±,ε |2, P ∗±,ε = |w±,ε |2, and the definition of ηi was extended to

i ∈ Z by periodicity. We claim that the right-hand side exactly corresponds to µ�τj (ηn)

([α]),

where µ�τj (ηn)

is a certain Bernoulli eigenmeasure for the iterated shift σn. The latter can be

seen as a simple shift on the symbol space � constructed from 3n symbols ε ∈ {0, . . . , 3n−1}:each ε is in one-to-one correspondence with a certain n-sequence ε0 . . . εn−1. Adapting theformalism of section 2.3.5 to this new symbol space, the Bernoulli measure µ�

τj (ηn)corresponds

to the following weight distributions P, P∗:

for ε ≡ ε0 . . . εn−1, Pε =n−1∏i=0

Pηn,j+i ,εi and P ∗ε =

n∏i=1

P ∗ηn,j−i ,εn−i . (6.10)

The measures µ�τj (ηn)

, j = 0, . . . , �η − 1, are related to one another through σ :

σ ∗ µ�τj (ηn) = |ληn,j |2 µ�τj+1(ηn). (6.11)

Finally, the semiclassical measure associated with the sequence (ψη,r )k=nk′→∞ is

µ�ηn = 1

Nηn

�η−1∑j=0

Cηn,j µ�τj (ηn)

, where Cηn,j =j−1∏m=0

|ληn,m |2�t

,

Nηn =n−1∑j=0

Cηn,j . (6.12)

This is a probability eigenmeasure of σ , with the decay rate �t . It only depends on the orbit{τ jηn, j = 0, . . . , �ηn − 1} and not on the choice of (r(k′)). This measure does not charge�\�′, so its push-forward is an eigenmeasure of B.

The proof of statement (ii) goes as follows: since ηn and η′n are not cyclically related, for

any j, j ′ ∈ Z, the weight distributions P and P′defining, respectively, the Bernoulli measures

µ�τj (ηn)

and µ�τj

′(η′n)

(see (6.10)) are different. As a result, these two measures are mutually

singular (see the end of section 2.3.5), and so are the two linear combinationsµ�ηn ,µ�η′n. Because

the weights P, P′

are different from (1, 0, . . . , 0) and (0, . . . , 1), the push-forwards µηn , µη′n

are also mutually singular. �By a standard density argument, we can exhibit semiclassical measures for arbitrary decay

rates in [|λ−|2, |λ+|2].


Corollary 2. Consider a real number t ∈ [0, 1] and a sequence of rationals (tp = (mp/np) ∈[0, 1])p→∞ converging to t when p → ∞. For each p, let ηp be a sequence with mp (+) andnp − mp (−) and µ�ηp the σ -eigenmeasure constructed above. Then any weak-∗ limit of the

sequence (µ�ηp )p→∞ is a semiclassical measure of (BN); it is a σ -eigenmeasure of the decay

rate �t = |λ1−t− λt+|2, and its push-forward is an eigenmeasure of B.

Let us call M�(�t) the family of semiclassical measures obtained this way, and M� =∪t∈[0,1]M

�(�t). We do not know whether the family M� exhausts the full set of semiclassicalmeasures for (BN). Still, we can address the third point in question 2 with respect to this family.

Proposition 9. The family of eigenmeasures M� does not contain the natural measureµ�nat = µ�rsym

. The same statement holds after push-forward on T2.

Proof. For any ε0 ∈ {0, 1, 2}, let us compute the weight of a measure µ�ηp ∈ M� on the

cylinder [ε0] (corresponding to the vertical rectangle Rε0 ). For the natural measure we haveµ�nat([ε0]) = 1/3 for ε0 = 0, 1, 2.

From (6.10), for any j ∈ Z one has µ�τj (ηp)

([ε0]) = Pηp,j ,ε0 . Combining this with (6.12),we get

µ�ηp ([ε0]) = C+ P+,ε0 + (1 − C+) P−,ε0 , where C+ = 1

Nηp

∑j : ηp,j=(+)

Cηp,j .

For any µ� ∈ M, the weights µ�([ε0]) will take the same form, for some C+ ∈ [0, 1]. Usingthe approximate expressions for the weights given in section 6.4.1, one finds that the conditionµ�([ε0]) = 1/3 cannot be satisfied simultaneously for ε0 = 0, 1, 2. �

7. Concluding remarks

The main result of this paper is the semiclassical connection between, on the one hand,eigenfunctions of a quantum open map (which mimic ‘resonance eigenfunctions’) and, onthe other hand, eigenmeasures of the classical open map.

We proved that, modulo some problems at the discontinuities of the classical map,semiclassical measures associated with long-living resonant are eigenmeasures of the classicaldynamics, and their decay rate is directly related with those of the corresponding resonanteigenstates (see theorem 1). This result, which basically derives from Egorov’s theorem,has been expressed in a quite general framework and applied to the specific example of theopen baker’s map. An analogue has been proven for the more realistic setting of Hamiltonianscattering [32, theorem 3].

Although the construction and classification of eigenmeasures with decay rates � < 1is quite easy, the classification of semiclassical measures among all possible eigenmeasuresremains largely open (see question 2). The solvable model provided by the Walsh-quantizedbaker provides some hints to these classification in the form of an explicit family ofsemiclassical measures, but we have no idea whether these results apply to more generalsystems, not even the ‘standard’ quantum open baker. Indeed, the high degeneracies of theWalsh–baker may be responsible for a nongeneric profusion of semiclassical measures for thismodel. Interestingly, the natural eigenmeasure does not seem to play a particular role at thequantum level.


A tempting way of constraining the set of semiclassical measures would be to adapt the‘entropic’ methods of [1] to open chaotic maps. A desirable output of these methods wouldbe, for instance, to forbid semiclassical measures from being of the pure point type describedin section 2.3.3.

Acknowledgments

MR thanks the Service de Physique Theorique for the hospitality in the spring of 2005, duringwhich this work was initiated. SN has been partially supported by the grant ANR-05-JCJC-0107-01 of the Agence Nationale de la Recherche. Both authors are grateful to J Keating,M Novaes and M Sieber for communicating their results concerning the Walsh-quantizedbaker before publication, and for interesting discussions. The authors also thank M Lebentalfor sharing with them her preliminary results on the open stadium billiard and the anonymousreferees for their stimulating comments.

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